Dynamic Programming: Manhattan Tourist Problem Lecture 17

1
Dynamic ProgrammingManhattan Tourist
ProblemLecture 17
2
ManhattanTouristProblem(MTP)
3
Manhattan Tourist Problem (MTP)
Imagine seeking a path (from source to sink) to
travel (only eastward and southward) with the
most number of attractions () in the Manhattan
grid
Source












Sink
4
Manhattan Tourist Problem (MTP)
Imagine seeking a path (from source to sink) to
travel (only eastward and southward) with the
most number of attractions () in the Manhattan
grid
Source












Sink
5
Manhattan Tourist Problem Formulation
Goal Find the longest path in a weighted grid.
Input A weighted grid G with two distinct
vertices, one labeled source and the other
labeled sink
Output A longest path in G from source to
sink
6
MTP An Example
0
1
2
3
4
j coordinate
source
3
2
4
0
9
5
3
0
0
1
0
4
3
2
2
3
2
4
13
1
1
6
5
4
2
0
7
3
4
19
15
2
i coordinate
4
5
2
4
1
3
3
0
2
3
20
3
8
5
6
5
sink
2
1
3
2
23
4
7
MTP Greedy Algorithm Is Not Optimal
1
2
5
source
3
10
5
5
2
5
1
3
5
3
1
4
2
3
promising start, but leads to bad choices!
5
0
2
0
22
0
0
0
sink
18
8
MTP Simple Recursive Program

• MT(n,m)
• if n0 or m0
• return MT(n,m)
• x ? MT(n-1,m)
• length of the edge from (n-
  1,m) to (n,m)
• y ? MT(n,m-1)
• length of the edge from
  (n,m-1) to (n,m)
• return maxx,y

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MTP Simple Recursive Program

• MT(n,m)
• x ? MT(n-1,m)
• length of the edge from (n-
  1,m) to (n,m)
• y ? MT(n,m-1)
• length of the edge from
  (n,m-1) to (n,m)
• return minx,y
• Whats wrong with this approach?

10
MTP Dynamic Programming
j
0
1
source
1
0
1
S0,1 1
i
5
1
5
S1,0 5

• Calculate optimal path score for each vertex in
  the graph
• Each vertexs score is the maximum of the prior
  vertices score plus the weight of the respective
  edge in between

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MTP Dynamic Programming (contd)
j
0
1
2
source
1
2
0
1
3
S0,2 3
i
5
3
-5
1
5
4
S1,1 4
3
2
8
S2,0 8
12
MTP Dynamic Programming (contd)
j
0
1
2
3
source
1
2
5
0
1
3
8
S3,0 8
i
5
10
3
1
-5
1
5
4
13
S1,2 13
3
5
-5
2
8
9
S2,1 9
0
3
8
S3,0 8
13
MTP Dynamic Programming (contd)
j
0
1
2
3
source
1
2
5
0
1
3
8
i
5
3
10
-5
-5
1
-5
1
5
4
13
8
S1,3 8
3
5
-3
3
-5
2
8
9
12
S2,2 12
0
0
0
3
8
9
S3,1 9
greedy alg. fails!
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MTP Dynamic Programming (contd)
j
0
1
2
3
source
1
2
5
0
1
3
8
i
5
3
10
-5
-5
1
-5
1
5
4
13
8
3
5
-3
2
3
3
-5
2
8
9
12
15
S2,3 15
0
0
-5
0
0
3
8
9
9
S3,2 9
15
MTP Dynamic Programming (contd)
j
0
1
2
3
source
1
2
5
0
1
3
8
Done!
i
5
3
10
-5
-5
1
-5
1
5
4
13
8
(showing all back-traces)
3
5
-3
2
3
3
-5
2
8
9
12
15
0
0
-5
1
0
0
0
3
8
9
9
16
S3,3 16
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MTP Recurrence
Computing the score for a point (i,j) by the
recurrence relation
The running time is n x m for a n by m grid (n
of rows, m of columns)
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Manhattan Is Not A Perfect Grid
What about diagonals?

• The score at point B is given by

18
Manhattan Is Not A Perfect Grid (contd)
Computing the score for point x is given by the
recurrence relation

• Predecessors (x) set of vertices that have
  edges leading to x
• The running time for a graph G(V, E)
  (V is the set of all vertices and E is
  the set of all edges) is O(E) since each
  edge is evaluated once

19
Traveling in the Grid

• The only hitch is that one must decide on the
  order in which visit the vertices
• By the time the vertex x is analyzed, the values
  sy for all its predecessors y should be computed
  otherwise we are in trouble.
• We need to traverse the vertices in some order
• Try to find such order for a directed cycle
• ???

20
DAG Directed Acyclic Graph

• Since Manhattan is not a perfect regular grid, we
  represent it as a DAG
• DAG for Dressing in the morning problem

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Topological Ordering

• A numbering of vertices of the graph is called
  topological ordering of the DAG if every edge of
  the DAG connects a vertex with a smaller label to
  a vertex with a larger label
• In other words, if vertices are positioned on a
  line in an increasing order of labels then all
  edges go from left to right.

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Topological ordering

• 2 different topological orderings of the DAG

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Longest Path in DAG Problem

• Goal Find a longest path between two vertices in
  a weighted DAG
• Input A weighted DAG G with source and sink
  vertices
• Output A longest path in G from source to sink

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Longest Path in DAG Dynamic Programming

• Suppose vertex v has indegree 3 and predecessors
  u1, u2, u3
• Longest path to v from source is
• In General
• sv maxu (su weight of edge from u to v)

su1 weight of edge from u1 to v su2 weight
of edge from u2 to v su3 weight of edge from
u3 to v
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Traversing the Manhattan Grid
a)
b)

• 3 different strategies
• a) Column by column
• b) Row by row
• c) Along diagonals

c)